Every graph is uniform-span $(2,2)$-choosable: Beyond the 1-2 conjecture

For a simple graph $G=(V,E)$, a \emph{proper total weighting} is a mapping $w: V\cup E\rightarrow \mathbb R$ such that for every edge $uv\in E$, $w(u)+\sum_{e\ni u}w(e)\neq w(v)+\sum_{e\ni v}w(e)$. The graph $G$ is said $(2,2)$-\emph{choosable} if, for any list assignment $L$ that assigns to each $z$ in $V\cup E$ a set $L(z)$ of two real numbers, there exists a {proper total weighting} $w$ with $w(z)\in L(z)$ for every $z\in V\cup E$. Wong and Zhu, and independently Przybyło and Woźniak conjectured that every simple graph is $(2,2)$-choosable. This conjecture remains open. For a set $\{a,b\}\subset \mathbb R$, its span is defined as $|b-a|$. We call a graph $G=(V,E)$ \emph{uniform-span} $(2,2)$-\emph{choosable} if, for any list assignment $L$ that assigns to every $z\in V\cup E$ a two-element list of a common span, there exists a {proper total weighting} respect to the assignment. In this paper, we present a novel lemma and perform comprehensive enhancements to our previous algorithm. These contributions enable us to prove that every graph is uniform-span $(2,2)$-choosable. This confirms the 1-2 conjecture in full generality, and provides supporting evidence for the $(2,2)$-choosable conjecture.